Solving variational inequalities with monotone operators on domains given by Linear Minimization Oracles

The standard algorithms for solving large-scale convex–concave saddle point problems, or, more generally, variational inequalities with monotone operators, are proximal type algorithms which at every iteration need to compute a prox-mapping , that is, to minimize over problem’s domain X the sum of a...

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Bibliographic Details
Published in:Mathematical programming Vol. 156; no. 1-2; pp. 221 - 256
Main Authors: Juditsky, Anatoli, Nemirovski, Arkadi
Format: Journal Article
Language:English
Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.03.2016
Springer Nature B.V
Springer Verlag
Subjects:
Accuracy
Algorithms
Calculus of Variations and Optimal Control; Optimization
Combinatorics
Computation
Computer programming
Convex analysis
Full Length Paper
Inequalities
Mathematical analysis
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
Mathematics of Computing
Minimization
Numerical Analysis
Operators (mathematics)
Optimization
Optimization and Control
Optimization techniques
Studies
Theoretical
United States > US
65K15
68T10
90C25
90C47
ISSN:0025-5610, 1436-4646
Online Access:Get full text
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Summary:The standard algorithms for solving large-scale convex–concave saddle point problems, or, more generally, variational inequalities with monotone operators, are proximal type algorithms which at every iteration need to compute a prox-mapping , that is, to minimize over problem’s domain X the sum of a linear form and the specific convex distance-generating function underlying the algorithms in question. (Relative) computational simplicity of prox-mappings, which is the standard requirement when implementing proximal algorithms, clearly implies the possibility to equip X with a relatively computationally cheap Linear Minimization Oracle (LMO) able to minimize over X linear forms. There are, however, important situations where a cheap LMO indeed is available, but where no proximal setup with easy-to-compute prox-mappings is known. This fact motivates our goal in this paper, which is to develop techniques for solving variational inequalities with monotone operators on domains given by LMO. The techniques we discuss can be viewed as a substantial extension of the proposed in Cox et al. (Math Program Ser B 148(1–2):143–180, 2014 ) method of nonsmooth convex minimization over an LMO-represented domain.
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ISSN:0025-5610
1436-4646
DOI:10.1007/s10107-015-0876-3